13 Whence those “rules of the game”?

Suppose that a maximal test performed at the time t1 yields the outcome u, and that we want to calculate the probability with which a maximal test performed at the later time t2 yields the outcome w. Further suppose that at some intermediate time t another maximal test is made, and that its possible outcomes are v1, v2, v3,… Let v be one of these values. Because a maximal test renders the outcomes of earlier measurements irrelevant, the joint probability p(w,v|u) with which the intermediate and final tests yield v and w, respectively, given the initial outcome u, is the product of two probabilities: the probability p(v|u) of v given u, and the probability p(w|v) of w given v. By Born’s rule, this is

p(w,v|u) = |<w|v> <v|u>|2,

where u, v, and w are unit vectors in the subspaces representing u, v, and w, respectively. To obtain the probability of w given u, regardless of the intermediate outcome, we must calculate this probability for all possible intermediate outcomes and add the results: